For laminar flow in a tube

Entrance flow is also accompanied by the growth of a boundary layer (Fig. 5b). As the boundary layer grows to fill the duct, the initially flat velocity profile is altered to yield the profile characteristic of steady-state flow in the downstream duct. For laminar flow in a tube, the distance required for the velocity at the center line to reach 99% of its asymptotic values is given by... 

For laminar flow in a tube, the flow rate is proportional to the fourth power of the radius and is inversely proportional to the length. 

Geometrically Similar Scaieups for Laminar Flow in a Tube... 

FIGURE 1.4 Velocity profile for laminar flow in a tube. 

As discussdd in Sec. 6.3, the velocity profile for laminar flow in a tube is parabolic. For turbulent flow it is much closer to plug flow, i.e., to a uniform velocity over the entire pipe cross section. Furthermore, as seen from Fig. 11.7, as the Reynolds number is increased, the velocity profile approaches closer and closer to plug flow. At the wall the turbulent eddies disappear so the shear stress at the wall for both laminar and turbulent flow of newtonian fluids is given byj dVJdy. Although it is ve difficult experimentally to... 

The way we have presented the one-dimensional dispersion model so far has been as a modification of the plug-flow model. Hence, u is treated as uniform across the tubular cross section. In fact, the general form of the model can be applied in numerous instances where this is not so. In such situations the dispersion coefficient D becomes a more complicated parameter describing the net effect of a number of different phenomena. This is nicely illustrated by the early work of Taylor [G.I. Taylor, Proc. Roy. Soc. (London), A219, 186 (1953) A223, 446 (1954) A224, 473 (1954)], a classical essay in fluid mechanics, on the combined contributions of the velocity profile and molecular diffusion to the residence-time distribution for laminar flow in a tube. 

Here Dg is the diffusivity of the salt, and the Reynolds number is defined by Eq. tl7-35bl with d = height of feed channel. This equation is similar to Eq. tl7-35aT but predicts a higher mass transfer coefficient. For laminar flow in a tube of length L and radius R with a bulk velocity U], the average mass transfer... 

Determine p for laminar flow in a tube. Solution Using Eq. (2.8-16),... 

The frictional pressure drop in both phases was calculated from the Hagen-Poiseuille equation for laminar flow in a tube. The interfacial pressure drop is described by the Bretherton s solution for the pressure drop over a single bubble in a capillary (Eq. 9.40) [63] ... 

Figure 3.2.8 Local and average Nusselt number (Nu Nu) for laminar flow in a tube (fully developed laminar flow, that is, without hydrodynamic entrance region, see Section 3.2.1.2.1).

For laminar flow in vertical tubes a series of charts developed by Pigford [Chem. Eng. Prog. Symp. Sen 17, 51, 79 (1955)] maybe used to predict values of... 

For laminar flow in a circular tube, the Leveque relationship is ... 

Maynes and Webb (2002) presented pressure drop, velocity and rms profile data for water flowing in a tube 0.705 mm in diameter, in the range of Re = 500-5,000. The velocity distribution in the cross-section of the tube was obtained using the molecular tagging velocimetry technique. The profiles for Re = 550,700,1,240, and 1,600 showed excellent agreement with laminar flow theory, as presented in Fig. 3.2. The profiles showed transitional behavior at Re > 2,100. In the range Re = 550-2,100 the Poiseuille number was Po = 64. 

For the most part of the experiments one can conclude that transition from laminar to turbulent flow in smooth and rough circular micro-tubes occurs at Reynolds numbers about RCcr = 2,000, corresponding to those in macro-channels. Note that other results were also reported. According to Yang et al. (2003) RCcr derived from the dependence of pressure drop on Reynolds number varied from RCcr = 1,200 to RCcr = 3,800. The lower value was obtained for the flow in a tube 4.01 mm in diameter, whereas the higher one was obtained for flow in a tube of 0.502mm diameter. These results look highly questionable since they contradict the data related to the flow in tubes of diameter d> mm. Actually, the 4.01 mm tube may be considered... 

Figure 13.7 illustrates stability regimes for the thermally initiated polymerization of styrene for laminar flow in a single tube. Design and operating variables... 

Daskopoulos, P., Lenhoff, A. M., Dispersion coefficient for laminar flow in curved tubes, AIChE J. 34, 12 (1988) 2052-2058. 

The solution to the problem of determining the wall shear rate for a non-Newtonian fluid in laminar flow in a tube relies on equation 2.6. 

Recall that the wall shear rate for a Newtonian fluid in laminar flow in a tube is equal to —8w/d,. In the case of a non-Newtonian fluid in laminar flow, the flow characteristic is no longer equal to the magnitude of the wall shear rate. However, the flow characteristic is still related uniquely to tw because the value of the integral, and hence the right hand side of equation 3.17, is determined by the value of tw. 

The quasi-steady laminar model is now employed to describe the heat transfer near the wall. Note that while the shear stress at the wall can be related easily to the pressure drop for the flow in a tube, it is more difficult to establish a relation between these two quantities for a packed or fluidized bed. However, while for the flow in a tube the dissipated energy is not uniform over the section... 

Thus, the pressure drop AP for laminar flow through a tube varies in proportion to the viscosity n, the average flow velocity v, and the tube length L, and in inverse proportion to the square of the tube diameter d. Since v is proportional to the total flow rate Q (m s ) and to d, C P should vary in proportion to ft, Q, L, and The principle of the capillary tube viscometer is based on this relationship. 

Solution Refer to Figure 4.2 and set Q = Q = Qout = 0.25m3/s, q = 0.75 m3/s, and V = I m3. Then t = 4 s for the overall system and 1 s for the once-through distribution. The differential distribution corresponding to laminar flow in a tube was found in Section 8.1.3. The corresponding washout function can be found using Equation (15.7). See also Section 15.2.2. The once-through washout function is... 

Fig. 5-3 Velocity profile for (a) laminar flow in a tube and (to) turbulent tube flow.

The developed velocity profile for turbulent flow in a tube will appear as shown in Fig. 5-15. A laminar sublayer, or film, occupies the space near the surface, while the central core of the flow is turbulent. To determine the heat transfer analytically for this situation, we require, as usual, a knowledge of the temperature distribution in the flow. To obtain this temperature distribution, the... 

Using the velocity distribution for developed laminar flow in a tube, derive an expression for the friction factor as defined by Eq. 5-112. 

Bergles, A. E., and R. R. Simonds Combined Forced and Free Convection for Laminar Flow in Horizontal Tubes with Uniform Heat Flux, Int. J. Heat Mass Transfer, vol. 14, p. 1989, 1971. 

For laminar airflow in a tube, when 8 approaches the tube radius, Poiseuille flow or a parabolic flow profile is fully developed. This is accomplished by the acceleration of the central portion of the flow. However, when Re exceeds a value lying somewhere between 104 and 106, the laminar boundary layer becomes so thick that it is no longer stable, and a turbulent boundary layer develops. 

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